A211855 -id:A211855 - cp's OEIS frontend

A211854 G.f. satisfies: A(x) = (1+xA(x)^2)(1+x^2A(x)^2)(1+x^3*A(x)^2).

1, 1, 3, 11, 42, 173, 746, 3321, 15155, 70516, 333282, 1595620, 7722036, 37715028, 185661034, 920244770, 4588778327, 23003827327, 115867080623, 586089365947, 2975978506450, 15163583668774, 77507719810688, 397320926569995, 2042152353063874

Offset: 0

Paul D. Hanna, Apr 22 2012

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 42*x^4 + 173*x^5 + 746*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 28*x^3 + 115*x^4 + 496*x^5 + 2211*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 84*x^3 + 391*x^4 + 1844*x^5 + 8800*x^6 +...
A(x)^6 = 1 + 6*x + 33*x^2 + 176*x^3 + 912*x^4 + 4674*x^5 + 23842*x^6 +...
where A(x) = 1 + x*(1+x+x^2)*A(x)^2 + x^3*(1+x+x^2)*A(x)^4 + x^6*A(x)^6.

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A182053, A211855.

{a(n)=local(A=1+x); for(i=1, n, A=(1+x*A^2)*(1+x^2*A^2)*(1+x^3*A^2)+x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

a(n) ~ s * sqrt((1 + 2*r + 4*r^3*s^2 + 5*r^4*s^2 + 6*r^5*s^4 + 3*r^2*(1 + s^2)) / (Pi*(1 + r + 6*r^3*s^2 + 6*r^4*s^2 + 15*r^5*s^4 + r^2*(1 + 6*s^2)))) / (2*n^(3/2)*r^n), where r = 0.1829152018931276962733907918487144062831105492965... and s = 1.828118673659452305128580127483211657533668751760... are real roots of the system of equations (1 + r*s^2)*(1 + r^2*s^2)*(1 + r^3*s^2) = s, 2*r*s*(1 + r + 2*r^3*s^2 + 2*r^4*s^2 + 3*r^5*s^4 + r^2*(1 + 2*s^2)) = 1. - Vaclav Kotesovec, Nov 22 2017

A182267 G.f. satisfies: A(x) = (1+xA(x))(1+x^2A(x)^2)(1+x^3*A(x)).

Original entry on oeis.org

1, 1, 2, 6, 16, 46, 140, 435, 1382, 4474, 14687, 48787, 163703, 554009, 1888794, 6481220, 22366415, 77575617, 270277602, 945480612, 3319582632, 11693824752, 41318554495, 146399071577, 520042511448, 1851657641932, 6607352892709, 23624965371264

Offset: 0

Paul D. Hanna, Apr 22 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 46*x^5 + 140*x^6 + 435*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 48*x^4 + 148*x^5 + 472*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 102*x^4 + 336*x^5 + 1124*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 185*x^4 + 648*x^5 + 2272*x^6 +...
where A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^3*(A(x) + A(x)^3) + x^4*A(x)^2 + x^5*A(x)^3 + x^6*A(x)^4.

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A182053, A211854, A211855.

{a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x^2*A^2)*(1+x^3*A)+x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

a(n) ~ sqrt(s*(1 + 2*r*s + 4*r^3*s + 5*r^4*s^2 + 6*r^5*s^3 + 3*r^2*(1 + s^2)) / (Pi*(1 + r^2 + 3*r*s + 3*r^3*s + 6*r^4*s^2))) / (2 * n^(3/2) * r^(n + 1/2)), where r = 0.2649675733882333627400730579639429790476557486165... and s = 2.383929237709193665917448862090331200952809331679... are roots of the system of equations (1 + r*s)*(1 + r^3*s)*(1 + r^2*s^2) = s, r*(1 + r^2 + 2*r*s + 2*r^3*s + 3*r^2*s^2 + 3*r^4*s^2 + 4*r^5*s^3) = 1. - Vaclav Kotesovec, Nov 18 2017

cp's OEIS Frontend

A211854 G.f. satisfies: A(x) = (1+xA(x)^2)(1+x^2A(x)^2)(1+x^3*A(x)^2).

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A182267 G.f. satisfies: A(x) = (1+xA(x))(1+x^2A(x)^2)(1+x^3*A(x)).

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cp's OEIS Frontend

A211854 G.f. satisfies: A(x) = (1+x*A(x)^2)*(1+x^2*A(x)^2)*(1+x^3*A(x)^2).

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PARI

Formula

A182267 G.f. satisfies: A(x) = (1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)).

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A211854 G.f. satisfies: A(x) = (1+xA(x)^2)(1+x^2A(x)^2)(1+x^3*A(x)^2).

A182267 G.f. satisfies: A(x) = (1+xA(x))(1+x^2A(x)^2)(1+x^3*A(x)).